$\dfrac{ -4e - 8f }{ 5 } = \dfrac{ e - 8g }{ 9 }$ Solve for $e$.
Multiply both sides by the left denominator. $\dfrac{ -4e - 8f }{ {5} } = \dfrac{ e - 8g }{ 9 }$ ${5} \cdot \dfrac{ -4e - 8f }{ {5} } = {5} \cdot \dfrac{ e - 8g }{ 9 }$ $-4e - 8f = {5} \cdot \dfrac { e - 8g }{ 9 }$ Multiply both sides by the right denominator. $-4e - 8f = 5 \cdot \dfrac{ e - 8g }{ {9} }$ ${9} \cdot \left( -4e - 8f \right) = {9} \cdot 5 \cdot \dfrac{ e - 8g }{ {9} }$ ${9} \cdot \left( -4e - 8f \right) = 5 \cdot \left( e - 8g \right)$ Distribute both sides ${9} \cdot \left( -4e - 8f \right) = {5} \cdot \left( e - 8g \right)$ $-{36}e - {72}f = {5}e - {40}g$ Combine $e$ terms on the left. $-{36e} - 72f = {5e} - 40g$ $-{41e} - 72f = -40g$ Move the $f$ term to the right. $-41e - {72f} = -40g$ $-41e = -40g + {72f}$ Isolate $e$ by dividing both sides by its coefficient. $-{41}e = -40g + 72f$ $e = \dfrac{ -40g + 72f }{ -{41} }$ Swap signs so the denominator isn't negative. $e = \dfrac{ {40}g - {72}f }{ {41} }$